Non-local games are studied in quantum information because they provide a simple way for proving the difference between the classical world and the quantum world. A non-local game is a cooperative game played by 2 or more players against a referee. The players cannot communicate but may share common random bits or a common quantum state. A referee sends an input $x_i$ to the $i^{th}$ player who then responds by sending an answer $a_i$ to the referee. The players win if the answers $a_i$ satisfy a condition that may depend on the inputs $x_i$. Typically, non-local games are studied in a framework where the referee picks the inputs from a known probability distribution. We initiate the study of non-local games in a worst-case scenario when the referee's probability distribution is unknown and study several non-local games in this scenario.
Non-local games are studied in quantum information because they provide a simple way for proving the difference between the classical world and the quantum world. A non-local game is a cooperative game played by 2 or more players against a referee. The players cannot communicate but may share common random bits or a common quantum state. A referee sends an input xi to the ith player who then responds by sending an answer ai to the referee. The players win if the answers ai satisfy a condition that may depend on the inputs xi. Typically, non-local games are studied in a framework where the referee picks the inputs from a known probability distribution. We initiate the study of non-local games in a worst-case scenario when the referee's probability distribution is unknown and study several non-local games in this scenario.
The notions of entanglement and nonlocality are among the most striking ingredients found in quantum information theory. One tool to better understand these notions is the model of nonlocal games; a mathematical framework that abstractly models a physical system. The simplest instance of a nonlocal game involves two players, Alice and Bob, who are not allowed to communicate with each other once the game has started and who play cooperatively against an adversary referred to as the referee. The focus of this thesis is a class of games called extended nonlocal games, of which nonlocal games are a subset. In an extended nonlocal game, the players initially share a tripartite state with the referee. In such games, the winning conditions for Alice and Bob may depend on outcomes of measurements made by the referee, on its part of the shared quantum state, in addition to Alice and Bob's answers to the questions sent by the referee. We build up the framework for extended nonlocal games and study their properties and how they relate to nonlocal games.
Abstract This paper considers the decidability of fully quantum nonlocal games with noisy maximally entangled states. Fully quantum nonlocal games are a generalization of nonlocal games, where both questions and answers are quantum and the referee performs a binary POVM measurement to decide whether they win the game after receiving the quantum answers from the players. The quantum value of a fully quantum nonlocal game is the supremum of the probability that they win the game, where the supremum is taken over all the possible entangled states shared between the players and all the valid quantum operations performed by the players. The seminal work MIP∗=RE~\cite{JNVWY'20,JNVWYuen'20} implies that it is undecidable to approximate the quantum value of a fully nonlocal game. This still holds even if the players are only allowed to share (arbitrarily many copies of) maximally entangled states. This paper investigates the case that the shared maximally entangled states are noisy. We prove that there is a computable upper bound on the copies of noisy maximally entangled states for the players to win a fully quantum nonlocal game with a probability arbitrarily close to the quantum value. This implies that it is decidable to approximate the quantum values of these games. Hence, the hardness of approximating the quantum value of a fully quantum nonlocal game is not robust against the noise in the shared states. This paper is built on the framework for the decidability of non-interactive simulations of joint distributions~\cite{7782969,doi:10.1137/1.9781611975031.174,Ghazi:2018:DRP:3235586.3235614} and generalizes the analogous result for nonlocal games in~\cite{qin2021nonlocal}. We extend the theory of Fourier analysis to the space of super-operators and prove several key results including an invariance principle and a dimension reduction for super-operators. These results are interesting in their own right and are believed to have further applications.
We study a generalization of non-local games—which we call extended non-local games —in which the players, Alice and Bob, initially share a tripartite quantum state with the referee. In such games, the winning conditions for Alice and Bob may depend on the outcomes of measurements made by the referee, on its part of the shared quantum state, in addition to Alice and Bob's answers to randomly selected questions. Our study of this class of games was inspired by the monogamy-of-entanglement games introduced by Tomamichel, Fehr, Kaniewski and Wehner, which they also generalize. We prove that a natural extension of the Navascués–Pironio–Acín hierarchy of semidefinite programmes converges to the optimal commuting measurement value of extended non-local games, and we prove two extensions of results of Tomamichel et al. concerning monogamy-of-entanglement games.
Nonlocality enables two parties to win specific games with probabilities strictly higher than allowed by any classical theory. Nevertheless, all known such examples consider games where the two parties have a common interest, since they jointly win or lose the game. The main question we ask here is whether the nonlocal feature of quantum mechanics can offer an advantage in a scenario where the two parties have conflicting interests. We answer this in the affirmative by presenting a simple conflicting interest game, where quantum strategies outperform classical ones. Moreover, we show that our game has a fair quantum equilibrium with higher payoffs for both players than in any fair classical equilibrium. Finally, we play the game using a commercial entangled photon source and demonstrate experimentally the quantum advantage.
Quantum entanglement, and the resulting peculiar non-classical correlations are one of the most counter-intuitive aspects of quantum mechanics.
The formalism of interactive games from computational complexity theory provides a useful framework in which to understand the power of entanglement. In an interactive game, a verifier interacts with a number of infinitely powerful provers who are allowed to share quantum entanglement but otherwise can't communicate. The ability of the verifier to extract useful information from the provers, whom he does not trust, provides an interesting measure of the ability of the provers to coordinate using their shared entanglement. Two particular interactive games we'll look at are Magic Square and 3SAT.
The Magic Square game is the iconic example of a game where two classical provers cannot perfectly coordinate their strategies using shared randomness, but quantum provers with shared entanglement can win with probability 1. We show that by adding an extra prover, we disallow perfect cheating. For 3SAT with three provers we also show that perfect cheating is not possible.
We then generalize the results for Magic Square and 3SAT by looking at non-commuting provers, a superset of entangled provers (communication is allowed, but operators applied by different provers must commute). Using this method, we obtain a generalized Tsirelson inequality that we apply to the Magic Square. Hence, we are able to give provably optimal strategies for the general Magic Square with n players. We also recover a similar result for 3SAT as with entangled provers, and we improve on it by showing that the gap is inverse exponential in the input size.
The no-signaling principle, which forbids faster than light communication, is a fundamental constraint on the non-local correlations resulting from quantum entanglement. However, no-signaling allows distributions that are more general than those arising from quantum mechanics. In particular, using a specific unit of general non-local correlation (the Popescu-Rohlich box), we show that there are classes that are equal to NEXP classically, but collapse to AM once such correlations are allowed. We also show that MIP where the verifier only looks at the XOR of the answers collapses to PSPACE. For the second approach, we show that by writing general non-local correlations as linear constraints, MIP is included in EXP under such correlations (vs NEXP classically).
We can extend these results to entangled quantum provers, by formulating an artificial MIP-like class built on a promise problem, that classically is equal to NEXP, but that also collapses to AM when quantum correlations are present.
The main objective of this paper is to provide illustrative examples of distributed computing problems for which it is possible to design tight lower bounds for quantum algorithms without having to manipulate concepts from quantum mechanics, at all. As a case study, we address the following class of 2-player problems. Alice (resp., Bob) receives a boolean x (resp., y) as input, and must return a boolean a (resp., b) as output. A game between Alice and Bob is defined by a pair (?, f) of boolean functions. The objective of Alice and Bob playing game (?, f) is, for every pair (x, y) of inputs, to output values a and b, respectively, satisfying ?(a, b) = f(x, y), in absence of any communication between the two players, but in presence of shared resources. The ability of the two players to solve the game then depends on the type of resources they share. It is known that, for the so-called CHSH game, i.e., for the game a ? b = x ? y, the ability for the players to use entangled quantum bits (qubits) helps. We show that, apart from the CHSH game, quantum correlations do not help, in the sense that, for every game not equivalent to the CHSH game, there exists a classical protocol (using shared randomness) whose probability of success is at least as large as the one of any protocol using quantum resources. This result holds for both worst case and average case analysis. It is achieved by considering a model stronger than quantum correlations, the non-signaling model, which subsumes quantum mechanics, but is far easier to handle.